Coherent control

Techniques to maintain quantum coherence
Coherent control is a quantum mechanics based method for controlling dynamic processes by light. The basic principle is to control quantum interference phenomena, typically by shaping the phase of laser pulses. The basic ideas have proliferated, finding vast application in spectroscopy mass spectra, quantum information processing, laser cooling, ultracold physics and more.

Coherent control aims to steer a quantum system from an initial state to a target state via an external field. For given initial and final (target) states, the coherent control is termed state to state control. A generalization is steering simultaneously an arbitrary set of initial pure states to an arbitrary set of final states i. e. controlling a unitary transformation. Such an application sets the foundation for a quantum gate operation. Controllability of a closed quantum system has been addressed by Tarn and Clark. Their theorem based in control theory states that for a finite dimensional, closed quantum system, the system is completely controllable, i. e. an arbitrary unitary transformation of the system can be realized by an appropriate application of the controls if the control operators and the unperturbed Hamiltonian generate the Lie algebra of all Hermitian operators. Complete controllability implies state to state controllability. The computational task of finding a control field for a particular state to state transformation is difficult and becomes more difficult with the increase in the size of the system. This task is in the class of hard inversion problems of high computational complexity. The algorithmic task of finding the field that generates a unitary transformation scales factorial more difficult with the size of the system. This is because a larger number of state to state control fields have to be found without interfering with the other control fields. It has been shown that solving general quantum optimal control problems is equivalent to solving Diophantine equations. It therefore follows from the negative answer to Hilbert's tenth problem that quantum optimal controllability is in general undecidable. Once constraints are imposed controllability can be degraded. For example, what is the minimum time required to achieve a control objective? This is termed the "quantum speed limit". The speed limit can be calculated by quantizing Ulam's control conjecture.

The constructive approach uses a set of predetermined control fields for which the control outcome can be inferred. The pump dump scheme which employs an auxiliary state to achieve complete state to state population transfer. One of the most prolific generic pulse shapes is a chirped pulse a pulse with a varying frequency in time.

Optimal control as applied in coherent control seeks the optimal control field for steering a quantum system to its objective. A local in time alternative method has been developed, where at each time step, the field is calculated to direct the state to the target. A related method has been called tracking